selection rules

selection rules [si′lek·shən ‚rülz]

(physics)

Rules summarizing the changes that must take place in the quantum numbers of a quantum-mechanical system for a transition between two states to take place with appreciable probability; transitions that do not agree with the selection rules are called forbidden and have considerably lower probability.

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Selection rules (physics)

General rules concerning the transitions which may occur between the states of a quantum-mechanical physical system. They derive in almost all cases from the symmetry properties of the states and of the interaction which gives rise to the transitions. The system may have a classical (nonquantum) counterpart, and in this case the selection rules may often be related to the classical conserved quantities. A first use of selection rules is in determining the symmetry classes of the states; but in a great variety of ways they may yield other information about the system and the conservation laws. See Quantum mechanics, Symmetry laws (physics)

For an isolated system the total angular momentum is a conserved quantity; this fact derives from a fundamental fact of nature, namely, that space is isotropic. Each state is then classifiable by angular momentum J and its z component M (= - J, -J + 1, …, +J). Angular momenta combine in a vectorial fashion. Thus, if the system makes a particle-emitting transition J₁,M₁ → J₂,M₂, the emitted particles must carry away angular momentum (j, μ), where j = j ₁ - j ₂. This implies that μ = M₁ - M₂ and that j takes on values J₁ - J₂, J₁ - J₂ + 1, …, (J₁ + J₂). Thus in transitions (J = 4 → J = 2) the possible j values comprise only 2, 3, 4, 5, 6, and, if it is also specified that M₁ - M₂ = ±4, only 4, 5, 6. Observe that J₂ is additive. See Angular momentum, Quantum numbers

Another fundamental symmetry, the parity, which determines the behavior of a system (or of its description) under inversion of the coordinate axes, is conserved by the strong and electromagnetic interactions, and gives a classification of systems as even (π = +1) or odd (π = -1). Under combination the parity combines multiplicatively. Thus, if the transition above is 4^± → 2^± , it follows that j^π = 2^-, …, 6^-, while 4^± → 2^± would give j^π = 2⁺, …, 6⁺. The angular momentum j may be a combination of intrinsic spin s and orbital angular momentum l . Scalar, pseudoscalar, vector, and pseudovector particles are respectively characterized by s^πs = 0⁺, 0^-, 1^-, 1⁺, where π_s is the “intrinsic” parity, while l always carries π_l = (-1)^l. See Parity (quantum mechanics), Spin (quantum mechanics)

The isospin symmetry of the elementary particles is almost conserved, being broken by electromagnetic and weak interactions. It is described by the group SU(2), of unimodular unitary transformations in two dimensions. Since the SU(2) algebra is identical with that of the angular momentum SO(3), isospin behaves like angular momentum with its three generators T replacing J .

The isospin group is a subgroup of SU(3) which defines a more complex fundamental symmetry of the elementary particles. Two of its eight generators commute, giving two additive quantum numbers, T_z and strangeness S^′ (or, equivalently, charge and hypercharge). The strangeness is conserved (ΔS^′ = 0) for strong and electromagnetic, but not for weak, interactions. The selection rules and combination laws for SU(3) and its many extensions, and the quark-structure underlying them, correlate an enormous amount of information and make many predictions about the elementary particles. See Baryon, Elementary particle, Meson, Quarks, Unitary symmetry

A great variety of other groups have been introduced to define relevant symmetries for atoms, molecules, nuclei, and elementary particles. They all have their own selection rules, representing one aspect of the symmetries of nature.